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Search: id:A076118
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| A076118 |
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a(n) = sum_k {n/2<=k<=n} k * (-1)^(n-k) * C(k,n-k). |
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+0
2
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| 0, 1, 1, -1, -3, -2, 2, 5, 3, -3, -7, -4, 4, 9, 5, -5, -11, -6, 6, 13, 7, -7, -15, -8, 8, 17, 9, -9, -19, -10, 10, 21, 11, -11, -23, -12, 12, 25, 13, -13, -27, -14, 14, 29, 15, -15, -31, -16, 16, 33, 17, -17, -35, -18, 18, 37, 19, -19, -39, -20, 20, 41, 21, -21, -43, -22, 22, 45, 23, -23, -47, -24, 24, 49, 25, -25, -51, -26, 26
(list; graph; listen)
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OFFSET
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0,5
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COMMENT
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Piecewise linear depending on residue modulo 6. Might be described as an inverse Catalan transform of the nonnegative integers.
Number of compositions of n consisting of at most two parts, all congruent to {0,2} mod 3 (offset 1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 10 2005
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FORMULA
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a(n) = ( 2n*sin((n+1/2)*pi/3) + sin(n*pi/3)/sin(pi/3) )/3. a(3n)=n*(-1)^n; a(3n+1)=(2n+1)*(-1)^n; a(3n+2)=(n+1)*(-1)^n.
a(n)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*(n-k)} - Paul Barry (pbarry(AT)wit.ie), Nov 12 2004
Euler transform of length 6 sequence [ 1, -2, -2, 0, 0, 2]. - Michael Somos Jul 14 2006
G.f.: x(1-x)/(1-x+x^2)^2 = x*(1-x^2)^2*(1-x^3)^2/((1-x)*(1-x^6)^2) . a(-1-n)=a(n) . - Michael Somos Jul 14 2006
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EXAMPLE
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a(10) =-5*1+6*15-7*35+8*28-9*9+10*1 =-5+90-245+224-81+10 =-7.
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PROGRAM
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(PARI) {a(n)=local(k=n%3); n=n\3; (-1)^n*((k>0)+n+(k==1)*n)} /* Michael Somos Jul 14 2006 */
(PARI) {a(n)=if(n<0, n=-1-n); polcoeff(x*(1-x)/(1-x+x^2)^2+x*O(x^n), n)} /* Michael Somos Jul 14 2006 */
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CROSSREFS
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Cf. A038608(n)=a(3n)=-a(3n-1).
Cf. A078028.
Sequence in context: A092743 A139420 A092895 this_sequence A058608 A112196 A021035
Adjacent sequences: A076115 A076116 A076117 this_sequence A076119 A076120 A076121
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KEYWORD
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sign
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AUTHOR
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Henry Bottomley (se16(AT)btinternet.com), Oct 31 2002
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